balanced fuzzy sets

8/6/2019 Balanced Fuzzy Sets

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Balanced fuzzy sets q

Wadysaw Homenda

Faculty of Mathematics and Information Science, Warsaw University of Technology,

pl. Politechniki 1, 00-660 Warszawa, Poland

Received 22 August 2004; received in revised form 29 November 2005; accepted 1 December 2005

Abstract

The paper presents a new approach to fuzzy sets and uncertain information based onan observation of asymmetry of classical fuzzy operators. Parallel is drawn betweensymmetry and negativity of uncertain information. The hypothesis is raised that classi-cal theory of fuzzy sets concentrates the whole negative information in the value 0 ofmembership function, what makes fuzzy operators asymmetrical. This hypothesiscould be seen as a contribution to a broad range discussion on unification of aggregat-ing operators and uncertain information processing rather than an opposition to otherapproaches. The new approach spreads negative information from the point 0 intothe interval [1,0] making scale and operators symmetrical. The balanced counterpartsof classical operators are introduced. Relations between classical and balanced opera-

tors are discussed and then developed to the hierarchies of balanced operators of higherranks. The relation between balanced norms, on one hand, and uninorms and null-norms, on the other, are quite close: balanced norms are related to equivalence classesof some equivalence relation build on linear dependency in the spaces of uninorms andnullnorms. It is worth to stress that this similarity is raised by two entirely differentapproaches to generalization of fuzzy operators. This observation validates the general-ized hierarchy of fuzzy operators to which both approaches converge. The discussion in

0020-0255/$ - see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.ins.2005.12.003

q

This work is supported under State Committee for Scientific Research Grant No. 3T11C00926,years 20042007.E-mail address: [email protected]

Information Sciences 176 (2006) 24672506www.elsevier.com/locate/ins
mailto:[email protected]:[email protected]


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this paper is aimed at presenting the idea and does not aspire to detailed exploration ofall related aspects of uncertainty and information processing.

2005 Elsevier Inc. All rights reserved.

Keywords: Fuzzy sets; Membership functions; Negative information; Triangular norms; Uninormsand nullnorms; Balanced norms

1. Introduction

The paper presents a new approach to fuzzy sets and uncertain information

based on an observation of asymmetry of classical fuzzy operators. Introduc-tory remarks and up to date background of the paper is presented in thissection. This presentation includes notes on works related to the merit of thispaper as well as fundamental definitions and properties of t-norms, t-conorms,uninorms and nullnorms. Motivation and aspiration to further discussion areoutlined in this section. Balanced operators are introduced in Section 2. Defi-nitions and basic properties of balanced t-conorms and balanced t-norms areincluded. Normal and weak forms of balanced t-norms are considered. Exam-ples of balanced t-conorms and t-norms supplement the section and the

balanced counterpart of classical maximum and minimum operators is setup. Section 3 brings up a discussion on similarities between balanced t-conormsand t-norms on one hand, and uninorms and nullnorms, on the other hand.These similarities prompt us to build a broader range of fuzzy operators.The hierarchies of balanced operators are constructed based on iterative oper-ators. Classical t-norms and t-conorms, balanced and t-norms, uninorms andnullnorms are considered as a basis of the hierarchy of balanced operators.

1.1. Preliminaries

A (crisp) set A in a universe Xcan be defined in the form of its characteristicfunction mA : X! {0,1} yielding the value 1 for elements belonging to the set Aand the value 0 for elements excluded from the set A. This representationallows for an easy definition of set operations: union, intersection and comple-ment. The max, min and complement to 1 (1), applied to characteristic func-tions of respective sets, express the crisp set operations.

A classical fuzzy set A in a universe X can be defined in terms of itsmembership function lA : X! [0, 1]. Membership functions, by analogy to

characteristic functions, define fuzzy connectives: union, intersection and com-plement. The definitions are expressed, as in the case of the crisp sets, by max,min and complement to 1 (1), cf. [41].

One can observe an asymmetry of the set of values of characteristic functionand membership functions: if the state of (certain) inclusion of an element is

2468 W. Homenda / Information Sciences 176 (2006) 24672506


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denoted by 1, the state of (certain)exclusion might be denoted by 1 ratherthan 0. This observation would be worthless since most of studies of fuzzy

set operators and all operators defined on crisp sets have been valued in theunit interval [0, 1]. Moreover, both scales: unipolar, with the unit interval[0,1], and bipolar, with the symmetric interval [1,1], are indistinguishablein the meaning of the linear mapping i(x) = 2x 1.

Yet, from psychological studies, it is known that human beings handle otherkinds of scale including the bipolar one modeled by the interval [1,1], cf. [16].It has been shown that the use of bipolar scales is of particular interest in sev-eral fields since they enable representation of the symmetry phenomena inhuman behavior: one can be faced with positive (gain, satisfaction, etc.) or neg-

ative (loss, dissatisfaction, etc.) quantities, but also with a kind of disinterest(does not matter, not interested in, etc.). For instance, someone either (1) likesto listen to music while reading an interesting novel or (2) does not like to listento music then or even (3) music is only a background not affecting him at all.These quantities could be interpreted in context of a relation between an ele-ment and a set as inclusion/exclusion/lack-of-inclusionexclusion-informationor so-called positive/negative/neutral information.

In economy the psychological attempt to decision-making process withuncertain premises overheads traditional models of customer behavior. The

pseudo-certainty effect is a concept from the prospect theory. It refers to peo-ples tendency to make risk-averse choices if the expected outcome is positive,but risk-seeking choices to avoid negative outcomes. Their choices can beaffected by simply reframing the descriptions of the outcomes without changingthe actual utility, cf. [24]. Aggregation of positive and negative premises leadsto implementation of a crisp decision. Modeling of such an attempt requiresprocessing of positive/neutral/negative information.

Assuming the unipolar scale [0,1], it is clear that a value close to 1 numer-ically expresses almost certain, strong, information about inclusion (inclusion

of element into set). In opposition, a question is asked whether a value closeto 0 expresses weak information about inclusion or rather strong informationabout exclusion. It would be expected that a value close to 0 declares strongexclusion rather than weak inclusion. Otherwise, assuming that numerical val-ues of membership function increase monotonically from negative throughneutral to positive information, membership function does not provide a mech-anism describing grades of exclusion. It is consistent to assume that the value1/2 expresses neutral information or lack of information, while values close to1 and close to 0 express strong, almost certain, information about inclusion or

about exclusion, respectively. This is a kind of harmonization of fuzzy operatorsatisfying expectation of symmetry. With regard to information aggregationthis expectation can be briefly outlined as a sort of the intuitive rule:

The preservation rule: aggregation operators should preserve strength ofaggregated information, i.e. to keep (or even to increase) strength of strong

W. Homenda / Information Sciences 176 (2006) 24672506 2469


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information, to keep (or even to increase) neutrality of neutral information andto make contradictory pieces of information neutral.

Considering aggregation of two pieces of information in the unipolar scale[0,1], the preservation rule could be expressed in the form of the following con-ditions, cf. Fig. 1:

1. if both pieces of information are strongly positive, i.e. their numerical valuesare close to 1 let us say 0.7 and 0.9 then the result is at least as strong asthe stronger piece of information, i.e. greater or equal to 0.9;

2. and vice versa, if both pieces of information are strongly negative, i.e. theirnumerical values are close to 0, e.g. 0.3 and 0.1 then the result is at least

as strong as the stronger piece of information, i.e. smaller or equal to 0.1;3. if both pieces of information are closely neutral, i.e. their numerical values

are close to 0.5 let us say 0.55 and 0.6 then the result is at least as weakas the weaker piece of information, i.e. equal to 0.55 or even smaller than0.55 (and, of course, greater than 0.5);

4. having contradictory pieces of information, i.e. strong positive and strongnegative ones take 0.7 and 0.3 it would be expected that the result ofaggregation is neutral, equal to 0.5.

The preservation rule has a symmetry property with the symmetry centerbeing neutral information and with negative/positive information as counter-parts of the symmetry. Of course, neutral information is expressed by thenumerical value 0.5. The asymmetry of unipolar scale of information modelingcan be removed by linear mapping i(x) = 2x 1 of the unit interval [0, 1] ontothe interval [1,1]. Moreover, such mapping preserves max and min functions

Fig. 1. Symmetry expectation in information aggregation.



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as numerical representations ofunion and intersection and gets opposite sign forthe complement, i.e. it is an isomorphic mapping.

However, the traditional aggregation operators do not satisfy the preserva-tion rule despite they are valued in the unit interval [0,1], the interval [ 1,1] orany other interval. For instance, union utilized as aggregation operator andmodeled by max operator valued in the unit interval [0,1] satisfies the first con-

dition of the preservation rule. Max function yields 0.9 as a result of aggrega-tion in the first case, but produces 0.3 instead of 0.1 in the second case. Strongt-conorm produces the value greater than 0.3 in the second case (cf. Section1.2.1 for description of t-norms and t-conorms). Thus, max operator and othert-conorms do not satisfy the preservation rule. The same problem is addressedwhen intersection modeled by min operator or any t-norm is used as aggrega-tion operator. A conclusion could be drawn that t-norms and t-conorms areoriented to positive information processing, cf. Fig. 2.

Therefore, the isomorphic mapping i(x) = 2x 1 applied in scale symmetri-

zation makes more confusion instead of solving inconsistency of these opera-tors. In fact, this isomorphism makes the scale symmetrical, but does not

0 11/2

t-norm

negative positiveneutral

0 11/2

t-conorm

0 1e=1/2

uninorm



0 1


nullnorm

e=1/2

Fig. 2. The behavior of aggregation operators.

0 1

0 1

dispersion of the crisp

positive

information

crisp information

crisp negative information

fuzzy positive information

Fig. 3. Crisp to fuzzy sets extension.



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change asymmetry of operators. These observations motivate us to draw thefollowing observation:

The asymmetry observation: the generalization of the crisp set to fuzzy setsrelied on spreading positive information that fit the crisp point {1} into theinterval (0,1], while negative information still remains fit into the crisp point{0}, cf. Fig. 3.

1.2. The background

So far most of the studies of aggregation operators have been conducted onthe unit interval [0,1] and they are consistent with the asymmetry observation.

The list of such operators includes t-norms, t-conorms, uninorms and null-norms, which are of special interest of further discussion in this paper. Theseoperators are briefly discussed in Sections 1.2.1 and 1.2.2. Yet, drawbacks ofoperators based on the unit interval [0,1] were observed in practice, and sonew operators were defined and investigated. Selected attempts are mentionedbelow.

In [36] OWA (ordered weighted averaging aggregation) operators were intro-duced. This family of new type operators were concerned with the problem ofaggregating multicriteria to form an overall decision function. In particular,

OWA operators have property of lying between the and, requiring all the cri-teria to be satisfied, and the or, requiring at least one criterion to be satisfied.In other words, OWA operators create a family of mean operators, cf. [37,38].

It is stated in [34] that classical fuzzy connectives are not symmetric undercomplement. Partitioning universe into two subsets, having comparable signif-icance, suffers from the asymmetry of fuzzy connectives. A class of operationson fuzzy sets called the symmetric summation is developed to avoid the asym-metry drawback of classical fuzzy operators.

The above attempts were based on the asymmetric model of fuzzy sets.

There are several attempts to explore negative information and the symmetryof information.

The twofold fuzzy set was derived in [10] by Dubois and Prade from thepossibility and necessity measures as a pair of fuzzy sets A+ and A, whereA+ represents the set of objects which possibly satisfy a non-vague property,and A represents the set of objects which necessarily satisfy the non-vagueproperty.

The so-called intuitionistic fuzzy sets an extension of fuzzy sets wereintroduced in [2]. This extension copes with a kind of negative information that

is attached to the classical fuzzy set as a separate component. This extra com-ponent deals with negative information creating a balance to dispersed positiveinformation of classical fuzzy sets. The system of intuitionistic fuzzy sets, how-ever, does not provide a tool to combine two separated types of informationbesides a simple condition the degree of indeterminacy.



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The concept of vague fuzzy sets presented in [17] relies on defining the gradeof membership function of an element x as the degree of truth tx and the degree

of falsity fx; both degrees included in the unit interval [0,1]. The unknown part1 tx fx is assumed to satisfy the inequalities 0 6 1 tx fx 6 1 whatmakes the concept of vague fuzzy sets is similar to that of intuitionistic fuzzysets.

In [16] the interval [1,1] is considered and endowed with an algebraicstructure like a ring. The motivation lies in decision-making, where scales which are symmetric with respect to 0 are needed to represent symmetry inthe behavior of the decision-maker. t-Conorms and t-norms are employedand relations to uninorms are done. The symmetric pseudo-addition and sym-

metric pseudo-multiplications in the interval [1,1] were defined and anattempt to build a ring is presented. It is shown that strict t-norms are ofimportance in algebraic structures created in the interval [1,1].

A rich survey of aggregation operators as well as a discussion on aspects ofinformation aggregation are presented in [8,31]. An interesting concept ofrescaling true/false concept of {0,1} values to the interval [1,1] is proposedand developed. This concept has a common idea with balanced triangularnorms: the unit square [0,1] [0,1] is reflected in the point (0,0) to definetrue/false values in the square [1,0] [1,0].

The very early medical expert system MYCIN, cf. [7], combines positive andnegative information by a somewhat ad hoc invented aggregation operator. In[12] it was shown that MYCIN aggregation operator is a particular case in aformal study on the aggregation of truth and falsity values.

Some other papers also consider values of membership function in the inter-val [1,1]. However, no negative meaning of information is suggested, cf.[5,19,28,42]. In [18] an extension of triangular norms to the interval [1,1] isintroduced. Yet, that extension is rather imprecisely defined and does not solvethe inconsistency between operators asymmetry and human symmetry

requirement.The preservation rule and symmetry expectations coincide with different

aspects of formal properties of aggregation operators, cf. [14,15]. The compen-sation and counterbalancement properties are interestingly close to the preser-vation rule, while reinforcement property outlines violation of symmetry oftriangular norms, see [13,40]. The studies on associative compensatory opera-tors are done in [25]. Interesting discussion on triangular norms application inthe context of information fusion could be found in [1,8,26,30,32]. Severalauthors, e.g. [29,33], have proposed a set of fundamental conditions defining

the aggregation operators. It is worthy of note that these basic definitionsare not necessarily compatible. In [31] a set of fundamental properties is pro-posed. This set of properties group all the precedent definitions under weakerconditions. Anyway, it is worth underlining that in spite of different perspectiveon information aggregation and, thus, different interpretations, properties of



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information aggregation methods, presented in this paper as well as outlined inother sources are amazingly compatible. This fact could be seen as a correct-

ness of a general trend leading to the creation of a unified attempt to informa-tion aggregation.

1.2.1. Triangular norms

Triangular norms were introduced as operators in probabilistic metricspaces, cf. [35]. The extended discussion on triangular norms is outlined in[27]. They were adopted as generalization of union and intersection of fuzzysets modeling and then applied in uncertain information processing.

Definition 1. Triangular norms, t-norms and t-conorms are mappingsp :[0,1] [0,1] ! [0, 1], where p stands for both t-norm and t-conorm,satisfying the following axioms:

A t-norm t (t-conorm s, respectively) is said to be strict if it is continuousand strictly monotoneaus (cf. [27]).

Additive generators of a strong t-norm (t-conorm) is said to be strong addi-tive generator.

Example 1. The following mappings are examples of t-norms and t-conorms:

1. Minimum and maximum, i.e. t(a, b) = min(a, b), s(a, b) = max(a, b).2. Product and probabilistic-sum, i.e. t(a, b) = a * b, s(a, b) = a + b a * b.3. Every strict t-norm t is represented by a function ft called an additive

generator:

tx;y gtftx fty

where ft is a continuous, strictly decreasing function, ft :[0,1] ! [0,+1], ft(1) = 0, limftx !

x!0

1,

gt is the inverse of ft.4. Every strict t-conorm is represented by a function fs called an additive

generator:

sx;y gsfsx fsy

1. p(a,p(b, c)) = p(p(a, b), c) Associativity2. p(a, b) = p(b, a) Commutativity3. p(a, b) 6p(c, d) if a 6 c and b 6 d Monotonicity4. t(1, a) = a for a 2 [0, 1] Boundary condition for t-norm t

s(0, a) = a for a 2 [0,1] Boundary conditions for t-conorm s



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where fs is a continuous, strictly increasing function,

fs :[0,1] ! [0,+1], fs(0) = 0, limftx !x!1 1, gs is the inverse of fs.

Definition 2. The t-norms and t-conorms are dual operations in the sense thatfor any given t-norm t, we have a dual t-conorm s defined by the De Morganformula s(a, b) = 1 t(1 a, 1 b) and vice versa, for any given t-conorms, we have a dual t-norm t defined by the De Morgan formula t(a, b) =1 s(1 a, 1 b).

Duality of t-norms and t-conorms raise duality of their properties. Forinstance, additive generators of strong t-norms and t-conorms satisfy the DeMorgan formula fs(x) = ft(1 x).

Note that the max/min and product/probabilistic-sum are pairs of dualt-norms and t-conorms.

The further discussion in this paper is concentrated on strict triangularnorms. Consideration of arbitrary triangular norms is a matter of furtherstudies.

Triangular norms utilized in data aggregation do not satisfy the preserva-

tion rule. This drawback is shown in a number of papers, e.g. [2,2022,34,36].To elaborate this defect in more detail, let us consider strong t-norm appliedto non-crisp arguments 0 < a, b < 1. The result value of t(a, b) is smallerthan the value of the smaller argument. It means that any strong t-norm doesnot satisfy the preservation rule, cf. Fig. 2. The same discussion concernst-conorms.

This characteristic can even be clearly outlined when convergence of aggre-gation of an infinite sequence of non-crisp equal data units is considered for thestrong t-norm t and t-conorm s:

lim ttt. . . ta; a . . .; a; a|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n times

!n!1

0 if 0 < a

1.2.2. Uninorms and nullnorms

Both the unit element 1 of a t-norm and the neutral element 0 of a t-con-orm are boundary points of the unit interval [0,1]. However, there are manyimportant operations whose neutral element is an interior point of the

underlying set. The fact that the first three axioms of triangular norms coin-cide, i.e. the only difference arises in the location of the neutral element, hasled to the introduction of a new class of binary operations closely related tot-norms and t-conorms the so-called uninorms, introduced in [39], also cf.[27].



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Definition 3. Uninorm is a mapping u :[0,1] [0,1] ! [0, 1] satisfying thefollowing axioms:

It is a direct consequence that a t-norm is a special uninorm with the neutralelement e = 1 and a t-conorm is a special uninorm with the neutral element

e = 0.Assuming that ue is a uninorm with the neutral element e and ifud is definedas ud(x,y) = 1 u(1 x, 1 y), then ud is a uninorm with the neutral element1 e. The uninorm ud is the dual uninorm to ue. This fact shows that uninormand its dual analogue differ quantitatively, but both are uninorms.

Assuming that u is a uninorm with identity e:

Uninorms generalize the concept of triangular norms. According to [11],assuming that u is a uninorm with the unit element e 2 (0,1), the mappings tuand su are the t-norm and t-conorm respectively:

tux;y uex; ey

eand sux;y

ue 1 ex; e 1 ey

1 e

or alternatively:

uux;y etux

e ;

y

e if x;y 2 0; eand

uux;y e 1 esux e

1 e;y e

1 e

if x;y 2 e; 1

The above results show that the domain of uninorm is split into the fourrectangular areas: the two squares determined by left-bottom and right-topvertexes {(0,0), (e, e)} and {(e, e),(1, 1)} respectively, on one hand, and thetwo remaining rectangles fitting up the unit square, on the other. Uninorm

restricted to the square determined by the vertexes {(0,0),(e, e)} is equal to asqueezed t-norm while the uninorm restricted to the square determined bythe vertexes {(e, e),(1,1)} is equal to a squeezed t-conorm. The values of a unin-orm in the two remaining rectangles of the unit square are between maximumand minimum values of its arguments, cf. Fig. 4.

1. u(a, u(b, c)) = u(u(a, b), c) Associativity2. u(a, b) = u(b, a) Commutativity3. u(a, b) 6 u(c, d) if a 6 c and b 6 d Monotonicity4. ($e 2 [0,1]) ("x 2 [0,1]) u(x, e) = x Boundary conditions,

e neutral element

1. u(a, 0) = 0 for all a 6 e, u(a, 1) = 1 for all aP e,2. x 6 u(x,y) 6 y for all x 6 e and e 6 y,3. Either u(0,1) = 0 or u(0, 1) = 1.



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Another variation of t-norms and t-conorms, modifying only the axiom ofboundary condition again, is introduced in [9]:

Definition 4. Nullnorm is a function v :[0,1] [0,1] ! [0,1] satisfying the fol-lowing axioms:

It is immediately clear that each nullnorm v satisfies the condition v(x, a) = afor all x 2 [0,1], cf. [27].

Nullnorms generalize the concept of triangular norms. Assuming that v is anullnorm with the annihilator a 2 (0,1), the mappings sv and tv are the t-conorm and t-norm respectively:

tvx;y vax; ay

a

and

svx;y va 1 ax; a 1 ay

1 aor alternatively:

vx;y atvx

a;y

a

if x;y 2 0; a

1. v(a, v(b, c)) = v(v(a, b), c) Associativity2. v(a, b) = v(b, a) Commutativity3. v(a, b) 6 v(c, d) if a 6 c and b 6 d Monotonicity4. ($a 2 [0, 1]) such that ("x 2 [0, a])

v(x, 0) = x and ("x 2 [a,1]) v(x, 0) = xBoundary conditions,a annihilator

1

1

0

squizzed

t-norm

squizzed

t-conorm

betweenmin

andmax

between

minand

max

e

e

The structure of uninorm

1

1

0

squizzed

t-conorm

squizzed

t-normconstant

value

a

constantvalue

a

a

a

The structure of nullnorm

Fig. 4. The structures of uninorms and nullnorms.



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and

vx;y a 1 asvx a

1 a ;

y a

1 a if x;y 2 a; 1For further discussion on uninorms and nullnorms see [3,4,27].

Likewise in the case of uninorms, the nullnorm domain is split into fourareas. Any nullnorm restricted to the square determined by the vertexes{(0,0),(a, a)} is equal to a squeezed t-conorm. Any nullnorm restricted to thesquare determined by the vertexes {(a, a),(1,1)} is equal to a squeezed t-norm.The values of any nullnorm in the two remaining rectangles of the unit squareare equal to the neutral element a, cf. Fig. 4.

1.2.3. Isomorphic transformation to the interval [1,1]The function h(x) = 2x 1 (and its inverse i(x) = (x + 1)/2), applied as iso-

morphic transformations between the spaces G(X) = [1,1]X = {ljl : X![1,1]} and F(X) = [0,1]X = {ljl : X! [0,1]}, gives the respective formulasfor t-norms and t-conorms based on the interval [1, 1]. To distinguish t-normsand t-conorms based on the unit interval [0,1] from those based on the interval[1,1], the later are dashed:

s;t : 1; 1 1; 1 ! 1; 1

sa; b hsh1a; h1b

ta; b hth1a; h1b

The isomorphic transformations between G(X) and F(X), h(x) = 2x 1 andi(x) = (x + 1)/2, applied to Example 1, give the following formulas

Negation is transformed to the function:

n : 1; 1 ! 1; 1 nA A

De Morgan formulas take the forms: sx;y ntnx; ny and fsx ftnx.

Remark 1. The isomorphic mapping h(x) = 2x 1 (and its inverse i(x) =(x + 1)/2) transforms uninorms and nullnorms into the interval [1,1] withvalues of the unit and neutral elements equal to e 2e 1 and a 2a 1,respectively. It is easily seen that the balanced isomorphic mappings:

1. ta; b mina; b sa; b maxa; b2. ta; b 1

2

a b a b 1 sa; b 1

2

a b a b 1

3. Every strong t-norm and t-conorm and their additive generatorsare drawn in similar way.



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hx

x e

efor x 2 0; e

x e

1 e for x 2 e; 1

8>: ix

ex 1

1 ex etransform uninorms (and nullnorm) to their symmetrized versions with unityand neutral elements equal to the value 0. Concluding there is no qualitativedifference between both spaces G(X) and F(X).

1.3. The aim

The paper aims at presenting an extension of classical fuzzy sets in themeaning of a dispersion of crisp negative information bunched in the point{0} into the interval [1,0]. This extension should not affect the structure ofclassical fuzzy sets based on the unit interval [0,1], cf. Fig. 5. Thus, classicalfuzzy sets will be immersed in a new space of balanced fuzzy sets. Since bothkinds of information positive and negative are equally important, it wouldbe expected that such an extension provides symmetry of positive/negativeinformation with center of symmetry placed in the point 0. This note can beformulated as the following:

The symmetry principle: the balanced extension of fuzzy sets relies onspreading negative information fitting the crisp point {0} of the fuzzy set intothe interval [1,0). The extension preserves properties of classical operators forpositive information. It provides the symmetry of positive/negative informa-tion with the center of symmetry placed in the point 0, cf. Figs. 5 and 6.

0 1crisp information

0 1-1 fuzzy positive informationfuzzy negative information


positive

information


negative

information

crisp negative information

Fig. 5. The method of balanced fuzzy sets extension.



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Classical aggregation operators are symmetrically reflected in the origin ofsystem coordinates. This reflection determines results of negative information

aggregation. Therefore, a balanced operator F is defined by the formulaF(x,y) = f(x, y) in the square [1,0] [1,0], where f is the respectiveclassical fuzzy operator. The interval [1,0] will include values yielded by bal-anced operator for negative values of arguments. Aggregation of differentkinds of information, i.e. pieces of positive and negative information, is uncon-strained and would be the subject of further research.

2. Balanced triangular norms

A balanced extension of triangular norms is discussed in this section. Thedefinition of balanced triangular norms, as introduced in [23], is derived fromthe definition of triangular norms. Contrary to the isomorphic rescaling, out-lined in Section 1.2.3, the extension method is solely based on spreading nega-tive information bunched in the point 0 into the interval [1, 0) and is consistentwith the symmetry principle. The discussion is based on strict triangular norms.

From now on the lower-case letters t, s, n, . . . will denote the classical oper-ators while the upper-case letters T, S, N, . . . will denote balanced operators.

2.1. Balanced negation and inverse operator

The symmetry principle brings the extension of the classical negationoperator to the whole interval [1,1], except for the point 0. The balanced

1

f classical

operator

F balanced

operator

-1

-1

classicalaggregation

operators

f:[0,3]x[0,3]->[0,3]

F(x,y)=f(x,y)

reflection

of classical

aggregation

operators:F(x,y)=-f(-x,-y)

unconstrained

area

of balanced

operators

F(x,y)

unconstrained

area

of balanced

operatorsF(x,y)

Fig. 6. The method of balanced operators extension.



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extension of the classical negation operator should be equal to 1 in the point 0.This is required due to the symmetry principle, which defines equality of clas-

sical and balanced operators in the unit interval [0,1]. On the other hand, dueto the nature of negation, the opposite value should be yielded, i.e.N(0) = N(0) = N(0). Only 0 satisfies this condition. This result is intui-tively justified: the state of ignorance cannot be reasonably inverted to the stateof full information. Moreover, it is confirmed by observation that even if theinverse of the state of ignorance might be assumed as a state of full informa-tion, there is no assertion to choose between inclusion and exclusion. As aresult, the solution assigning the value 0 to the argument 0 is reasonable inpractice. This operator should be identical with classical negation operator

n(A) = 1 A on the unit interval (0, 1] and symmetrical on the interval[1,0), as it is noted above. Such an operator is called the inverse operator:

Ix

1 x x > 0

1 x where x < 0

0 x 0

8>>>:

9>>=>>;

or alternatively:

Ix

1 x x > 0

N1 Nx where x < 0

0 x 0

8>>>:9>>=>>;

Note: the inverse value of the state of ignorance could be alternativelydefined as undefined.

The inverse operator cannot be used as a balanced negation operator. Thebalanced negation should assign the opposite value to the given argument:the balanced negation operator should transform the state of inclusion into

the state of exclusion, i.e. N(a) = a. Note that such a condition is satisfiedby the isomorphic rescaling of the classical negation operator to the interval[1,1], denoted n, cf. Section 1.2.3. Yet, the identity of both operators isaccidental.

Thus, the following formula defines the balanced negation operator:

N : 1; 1 ! 1; 1 NA A

2.2. Balanced triangular norms

Balanced t-conorm and balanced t-norm in normal form. The definition of bal-anced t-conorms and balanced t-norm in the normal form is drawn from the setof axioms of triangular norms by extending the domain of triangular normsand providing the symmetry axiom.



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Definition 5. The balanced t-norm and t-conorm are mappings P: [1,1] [1,1] ! [1,1], where Pstands for both the balanced t-norm and balanced t-

conorm, satisfying the following axioms:

Conclusion 1. Definitions of the balanced t-norm and the balanced t-conormrestricted to the unit square [0,1] [0,1] are equivalent to the classical t-normand classical t-conorm, respectively.

Conclusion 2. The balanced t-norm and the balanced t-conorm restricted tothe square [1,0] [1,0] are isomorphic with the classical t-conorm and clas-

sical t-norm, respectively.

Conclusion 3. The balanced t-norm vanishes in even quarters of the domain,i.e. in the squares [0,1] [1,0] and [1,0] [0,1]

The above conclusions are obvious.Weak form of balanced t-norm. The balanced t-norm in the weak form is

drawn from the balanced t-norm in the normal form by weakening axiomsof associativity and monotonicity.

The normal form of the balanced t-norm yields zero for arguments of oppo-site signs. This property makes the balanced t-norm to be insensitive in half ofits domain. The insensitivity drawback is forced by the axioms of monotonicityand boundary conditions. Consequently, modification of one of these axiomsmay lead to elimination of insensitivity. The boundary condition axiom shouldrather not be changed due to desired consistency with the classical t-norm. Con-sequently, the property of monotonicity would be a subject of possibleadjustment. However, twisted monotonicity would produce perturbation ofassociativity. Therefore, the weak form of the balanced t-norm would be built

on the basis of the balanced t-norm in normal form with diluted monotonicityand associativity. On the other hand, both the preservation of classical fuzzy setproperties and the symmetry of balanced fuzzy sets are required. Yet, these twoprerequisites need satisfaction of all assumptions of classical theory for argu-ments of the same sign. As a result, modification of any property is possible only

1. P(a, P(b, c)) = P(P(a, b), c) Associativity2. P(a, b) = P(b, a) Commutativity3. P(a, b) 6 P(c, d) for a 6 c and b 6 d Monotonicity4. S(0, a) = a for a 2 [0, 1] Boundary condition for balanced

t-conorm ST(1, a) = a for a 2 [0, 1] Boundary conditions for balanced

t-norm T

5. P(x,y) = N(P(N(x), P(N(y)))) Symmetry



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in the case of arguments of opposite signs. Finally, due to unaffected boundaryconditions, the monotonicity property could be modified for arguments of

opposite signs. The diluted global property of monotonicity breaks the globalproperty of associativity. Conversely, tolerable variations of associativityshould be minimized. The possible limit could be raised by the value of the bal-anced t-norm for absolute values of arguments. The following set of axiomscompletes the definition of the balanced t-norm in the weak form:

Definition 6. The balanced t-norm in the weak form T is a function satisfyingthe following set of axioms:

Note: The symbol of absolute value used in the first axiom defines itsmeaning.

Conclusion 4. The definition of the balanced t-norm in the weakform restricted to the unit square [0,1] [0,1] is equivalent to the classical t-norm.

Proof. Satisfaction of commutativity, monotonicity and boundary conditionsis a direct conclusion from the definition of the balanced t-norm in the weakform because the symmetry axiom does not affect values of the balancedt-norm for non-negative arguments.

Finally, for non-negative arguments a, b, c the balanced t-norm in the weakform produces non-negative values. For such arguments the balanced t-norm isassociative. This is a direct conclusion from the above remark concerningmonotonicity and boundary conditions. Consequently, the left-hand side of thesemi-association formula becomes equal to the right-hand side of the formula:min(jT(a, T(b, c))j, jT(T(a, b), c)j) = T(a, T(b, c)) = T(T(a, b), c) and this ends theproof. h

Conclusion 5. The balanced t-norm in the weak form restricted to the square[1,0] [1,0] is isomorphic with the classical t-conorm.

Proof. This is a direct conclusion from the symmetry axiom and Conclusion4. h

1. minjTa; Tb; cj; jTTa; b; cj 6 jTa; Tb; cj;jTTa; b; cj 6 TTjaj; jbj; jcj Tjaj; Tjbj; jcj

Semi-associativity

2. T(a, b) = T(b, a) Commutativity3. T(a, b) 6 T(c, d) for 0 6 a 6 c, 0 6 b 6 d Semi-monotonicity4. T(1, a) = a for aP 0 Boundary conditions5. T(x,y) = N(T(N(x), N(y))) Symmetry



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Duality and consistency of balanced t-conorms and t-norms. Balanced t-normsand balanced t-conorms preserve the classical De Morgan association formula

in the unit square [0,1] [0,1]. Symmetry of balanced operators guaranteesthat this formula is also satisfied in the square [1,0] [1,0]. However, thelack of strict dependencies between the balanced t-conorm and the balancedt-norm in the normal form as well as diluted associativity and monotonicityof the balanced t-norm in the weak form affect the De Morgan associationformula in even quarters of the square [1,1] [1,1]. Due to the symmetryprinciple the balanced counterpart of the classical De Morgan associationformula should preserve the classical formula in odd quarters of the square[1,1] [1,1]. Moreover, it would be desired that the balanced De Morgan

formula keeps some dependency between the balanced t-conorm and the bal-anced t-norm in even quarters of the square [1,1] [1,1]. The following def-inition formalizes the above remarks on the balanced De Morgan associationformula.

Definition 7. The balanced t-norm T and the balanced t-conorm S are dual ifthey satisfy the following balanced De Morgan formulas:

Due to the symmetry of balanced operators the second condition is directlyextended on the square [1,0] [1,0].

The classical De Morgan formula defines unique dual operators for a givent-norm or t-conorm. It is clear that duality of balanced operators is not aunique relation. More than one dual balanced operator exists for a given bal-anced t-norm or a given balanced t-conorm.

The basic set operations and fuzzy set operations intersection and union

are modeled numerically as t-norms and t-conorms. The intuitive and formalcondition says that the intersection of classical fuzzy sets brings less informa-tion than their union, i.e.:

tx;y 6 sx;y for x;y 2 0; 1

This condition is satisfied by balanced norms having arguments of the samesign, i.e. the inequality 0 6 T(x,y) 6 S(x,y) is satisfied in the unit square[0,1] [0, 1] and, due to the symmetry principle, the inequality S(x,y) 6T(x,y) 6 0 is satisfied in the square [1,0] [1,0]. It is reasonable and desiredto extend this condition to the whole domain of balanced operators.

Definition 8. The balanced t-norm Tand the t-conorm Sare consistent if theysatisfy the inequality:

jTx;yj 6 jSx;yj for x;y 2 1; 1

1. jT(x,y)j 6 jI(S(I(x), I(y)))j for (x,y) 2 [1,1] [1,1]

2. T(x,y) = I(S(I(x), I(y))) for (x,y) 2 [0,1] [0,1]



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This inequality is trivial for any pair of the balanced t-conorm andthe balanced t-norm in the normal form. Any balanced t-conorm and bal-

anced t-norm in the normal form are obviously consistent. However, notevery balanced t-conorm and balanced t-norm in the weak form areconsistent.

2.3. Examples of balanced triangular norms

Examples of balanced t-conorms and balanced t-norms are based on anintuitive extension of additive generators of the classical t-conorm. Balancedt-norms in the normal form and in the weak form are derived from the bal-anced t-conorm. Then, balanced counterparts of maximum and minimumare built as the limit of a convergent sequence of strong balanced t-conormsand t-norms.

Balanced t-conorm. The additive generator of a strong t-conorm could beeasily extended in the spirit of the symmetry principle by applying the DeMorgan formula. Thus, a balanced t-conorm may be defined using respectiveformula for classical additive generators. Then, applying the analogy ofthe De Morgan formula, cf. Definition 2, the balanced t-conorm could bedefined.

Proposition 1. The function S : [1,1] [1,1] ! [1,1], S(x,y) = gs(fs(x) +fs(y)) is balanced t-conorm, where

fs is an additive generator satisfying the following properties:

fs : [1,1] ! [1, 1], it is continuous, it is strictly increasing, limfsx !

x!11,

it is odd, i.e. fs(x) = fs(x), f s : [1,1] ! [1, 1];

gs is the inverse function of fs.

This definition satisfies the set of axioms of the classical t-conorm, whatassures that the balanced t-conorm is equal to the classical t-conorm in the unitsquare [0, 1] [0, 1]. Obviously, the symmetry principle is satisfied in thesquares [1,0] [1,0] and [0,1] [0,1]. Moreover, the symmetry conditionS(x, y) = S(x,y) is satisfied in the domain [1,1] [1,1] {(1,1),(1, 1)}. The two vertexes {(1,1),(1, 1)} of the square [1,1] [1,1] are

points of irregularity of the balanced t-conorm defined by the strong additivegenerator. The value of the balanced t-conorm should be explicitly defined inthese points. Both irregularity points represent pieces of certain, but contradic-tory, information. Intuitively, aggregation of such pieces of information wouldbe defined as yielding no information (yielding the ignorance state). However,



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such a definition leads to a conflict with the desired associativity property:S(1, S(1, 1)) = 15 0 = S(S(1,1), 1). Thus, in this paper values of a balanced

t-conorm are assumed to be undefined in the corners {(1,1),(1,1)}. Valuesof a balanced t-conorm in these corners could also be assumed to be either 1 or

1, cf. [12].

Example 2. The following function can be considered as an examples of thestrong additive generator of the balanced t-conorms, S:

fs : 1; 1 ! 1;1 fsx x

1 jxj

fs : 1; 1 ! 1;1 fsx tanh1

x

Balanced t-norm in normal form. An additive generator of a balanced t-normcould be constructed based on an additive generator of the classical t-norm andin the spirit of the symmetry principle by analogy to the construction of anadditive generator of the balanced t-conorm. Such an extended additive gener-ator will have an irregularity point 0. According to the symmetry principle, thevalue of an additive generator in the point 0 should either be equal to 0 orshould be undefined. These assumptions lead to the following conditions:

ft is the generic function that satisfies the following properties: ft : [1,1] ! [1, 1], it is continuous on the intervals [1,0) [ (0,1], it is strictly decreasing on the intervals, [1,0) [ (0,1], it is odd, i.e. ft(x) = ft(x), ft : [1,1] ! [1, 1], ft has an irregularity in the point 0, the value in this point is described as

ft(0) = 0;

gt is the inverse function of ft.

The formula T(x,y) = gt(ft(x) + ft(y)) would define the balanced t-norm.This description satisfies the set of axioms of the classical t-norm and the sym-metry principle in the squares [1,0] [1,0] and [0,1] [0,1].

Alternatively, using the balanced De Morgan formula and having in mindthe monotonicity property of the balanced t-norm in the normal form, wecome to the following:

Proposition 2. Balanced t-norm in normal form, T, is defined by the following

formula:

Tx;y ISIx;Iy x;y : and

0 elsewhere

(

as shown in Fig. 7.



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Moreover, interpretation of this definition is intuitively straightforward. The

balanced t-norm in the normal form is determined by its dual t-conorm andsymmetry axiom in odd quarters of the square [1,1] [1,1]. In contrast,no reasonable information could be yielded by the balanced t-norm Tfor argu-ments of different signs. Due to its regularity, this form of balanced t-norm issaid to be normal. It is clear that both the balanced t-conorm S and the bal-anced t-norm T, as defined here, are dual and are consistent.

Balanced t-norm in weak form. The formula T(x,y) = gt(ft(x) + ft(y)) isunstable for arguments of different signs. Having absolute values of argumentsclose to each other and close to 0, it gives the result close to 1, e.g.

t(0.1, 0.1001) = 0.9. Moreover, the desired consistency inequality does nothold. Thus, the formula T(x,y) = gt(ft(x) + ft(y)) cannot be applied as a defini-tion of the balanced t-norm in the whole domain [1,1] [1,1]. Concluding,the balanced t-norm cannot be defined by analogy to the classical De Morganformula: T(x,y) = I(S(I(x), I(y))). Instead, two forms of balanced t-norms inweak forms are presented. It is assumed that an additive generator of the bal-anced t-conorm Sapplied in both definitions is an extension of a strict additivegenerator of a t-conorm.

Proposition 3. A balanced t-norm is said to be in the weak inertia form if itsatisfies the inertia assumption: the function TXp x Tx;p and the functionTYpx Tp;x are both continuous and increasing for a given value of theparameter p, p5 0, and for values of the argument x close to 0. However, this

solution brings a non-intuitive feature the produced value of the balanced

(*)

(**)

(***)

(***)

(****)

(****)

0-1

-1

1

1

Y

X

(*****)

Fig. 7. The domain of balanced triangular norm.



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t-norm, when applied to arguments of different signs, inherits its sign from the

weaker argument, i.e. argument of the smaller absolute value. The following

formula defines a balanced t-norm in the weak inertia form:

Tx;y

ISIx;Iy x;y : and

ISSIjxj;Ijyj;IjSx;yj x;y :


0 x;y :

8>>>>>:

as shown in Fig. 7.

Note: The symbol of absolute value is a shortcut for the following

transformation:

jxj x for xP 0

Nx for x < 0

Proposition 4. The balanced t-norm is said to be in the weak intuitive form if its

value computed for arguments of different signs inherits the sign from its stronger

argument. This means that the function TXp x Tx;p and the functionTYpx Tp;x are both locally continuous for a given value of the parameter p

and for values of the argument x close to 0. Both functions TXp x Tx;p andTYpx Tp;x have local extreme for argument equal to 0, i.e. they have localminimum for positive values p and local maximum for negative values p. The

following formula defines a balanced t-norm in the weak intuitive form:

Tx;y

ISIx;Iy x;y : and



0 x;y :

8>>>>>:as shown in Fig. 7.

Contour plots of examples of balanced triangular norms are shown inFig. 8. These norms are based on the additive generator fs(x) = x/(1 jxj).

Conclusion 6. Assuming that an additive generator of the balanced t-conormS, applied in Propositions 3 and 4, is an extension of a strict t-conorm additivegenerator, then the balanced t-norm in the weak form is continuous in thewhole domain except for the left-upper and right-bottom vertexes, i.e. it is

continuous in the set [1,1] [1,1] {(1,1),(1, 1)}. The balanced t-normin the weak form is assumed to be undefined in the irregularity corners{(1,1),(1, 1)}.

This conclusion is obvious.



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Conclusion 7. The balanced t-norm defined in Propositions 3 and 4 satisfy thebalanced De Morgan association formula.

The proof is given in Appendix A.

Conclusion 8. The balanced t-norms defined in Propositions 3 and 4 areconsistent:

jTx;yj 6 jSx;yj for x;y 2 1; 1 1; 1

The proof is given in Appendix A.

Fig. 8. Examples of balanced triangular norms based on the additive generator fS(x) = x/(1 jxj).



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Balanced counterparts of max and min. Since the strong t-norm and t-conormcan approximate basic triangular norms min and max, then this observation can

be extended to the whole domain of balanced triangular norms. Letfn(x) = sign(x) * (jxj

n/(1 jxjn)) be a sequence of additive generators of bal-anced t-conorms defined in Proposition 1. It can be shown that the balancedt-conorms Sn can approximate t-conorm max in the unit square [0,1] [0,1]to arbitrary accuracy, i.e. it can be shown that (cf. Conclusion 9 in Appendix A):

lim Sx;y !n!1

maxx;y for any values 0 6 x 6 1 and 0 6 y6 1

This limit formula could be used as a basis of a definition of the balancedt-conorm max, i.e. it could be extended to the whole domain [1,1] [1,1]

as follow:S maxx;y lim

n!1Snx;y for x;y 2 1; 1 1; 1

fx;y : x 6 0 and x 6 yg

The limit formula, if extended to the whole domain, will break associativity,cf.

S maxS max0:3;0:5; 0:5 0 6 0:3

S max0:3; S max0:5; 0:5

Assuming that S_max is undefined for x = y and x5 0 we get associativityproperty.

The final formula for S_max looks as follow:

S maxx;y

maxx;y

NmaxNx;Ny

0

undefined

8>>>>>:

for

x y> 0

x y< 0

x 0 y

otherwise

8>>>>>:

and for x;y2 1;11;1

Note: The values ofS_max(x, x) could be defined alternatively as x or x,cf. [12].

By analogy, the balanced t-norm min in normal and weak forms could bedefined by the limit formula:

T minx;y limn!1

Tnx;y for x;y 2 1; 1 1; 1

where Tn are balanced t-norms generated by additive generators dual to fnaccording to Propositions 24. The normal and weak inertia forms of the bal-

anced t-norm T_min are finally described as:

T minx;y

minx;y

NminNx;Ny

0

8>: for

xP 0;yP 0

x6 0;y6 0

otherwise

8>: normal form



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T minx;y

minx;y

NminNx;Ny

NminNx;yNminx;Ny

minNx;y

minx;Ny

0

8>>>>>>>>>>>>>>>>>>>>>:

for

xP 0;yP 0

x6 0;y6 0

x< 0;y> 0;x y> 0x> 0;y< 0;x y> 0

x< 0;y> 0;x y< 0

x> 0;y< 0;x y< 0

otherwise

8>>>>>>>>>>>>>>>>>>>>>:

weak inertia form

Example 3. Let us consider a simple decision-making process in real econom-ical environment, i.e. with uncertain premises. Having numerical representation

of the set of six premises with values in the unit interval [0, 1], let us assume thatvalues smaller than 0.5 indicate negative decision while values greater than 0.5suggest positive decision, while 0.5 is the neutral value. Let us consider twodifferent sets of numerical values of premises. The numerical values of thefirst set are equal to l1 = 0.6, l2 = 0.6, l3 = 0.6, l4 = 0.6, l5 = 0.6, l6 = 0.4while the values of the second set are equal to m1 = 0.6, m2 = 0.4, m3 = 0.4,m4 = 0.4, m5 = 0.4, m6 = 0.4. Classical fuzzy connectives employed as aggregatorsof premises do not distinguish between these two sets: max and min operatorsproduce the same value for both sets 0.6 and 0.4 respectively. Employing the

strong t-conorm based on the additive operator f(x) = x/(1 x), we get thevalues 0.89 and 0.83 for premises li and mi respectively. In the case of a dualt-norm we get the respective values equal to 0.11 and 0.17. The isomorphictransformation to the interval [1,1] in terms of Section 1.2.3 gives thefollowing values of strong triangular norms: 0.78, 0.66 and 0.22, 0.34.Therefore, besides small quantitative differences between aggregated numericalresults no qualitative indication is given with regard to the decision.

Employing the balanced modeling we get the transformed values of premises

equal to l1 = 0.2, l2 = 0.2, l3 = 0.2, l4 = 0.2, l5 = 0.2, l6 = 0.2 and m1 = 0.2,m2 = 0.2, m3 = 0.2, m4 = 0.2, m5 = 0.2, m6 = 0.2. In this modeling thepositive value of aggregated premises indicate the positive decision and the neg-ative value the negative decision. The balanced t-conorm based on the addi-tive operator f(x) = x/(1 jxj) produces the values 0.85 and 0.85 for premisesli and mi, respectively, while the dual balanced t-norm in normal form producesthe values 0.27 and 0.27, respectively. In the case of the balanced modeling aclear indication is given with regard to the decision.

3. Hierarchies of fuzzy operators

Amazingly, balanced triangular norms as well as uninorms and nullnormsare similar products of two different attempts to fuzzy sets. Detailed properties



26/40

of balanced triangular norms, on one hand, and uninorms and nullnorms, onthe other, might differ. Despite this, the general meaning of balanced triangular

norms and of uninorms and nullnorms are the same in the sense of isomorphicmapping between them.

3.1. Balanced triangular norms versus uninorms and nullnorms

Balanced t-conorms, as defined in Definition 5, are special cases of uninormsin the sense of the isomorphic mappings defined in Remark 1.

The definition of the balanced t-conorm includes the symmetry axiom inaddition to other axioms that are common for uninorms and balanced t-con-

orms: associativity, commutativity, monotonicity and boundary conditions.This extra restriction, i.e. the symmetry axiom, causes that not every uninormis isomorphic with a balanced t-conorm while every balanced t-conorm is iso-morphic with a uninorm. Precisely, every balanced t-conorm is isomorphicwith a set of uninorms that have upper-right part isomorphic with the t-con-orm, that are symmetrical and that may have the unit element different. Sym-metry of the uninorm means that both parts of it: t-norm and t-conorm aredual. Of course, any two uninorms of such a set are isomorphic in the senseof the balanced isomorphism defined in Remark 1. Two sets of uninorms

related to any two balanced t-conorms are disjoint assuming that the respectivebalanced t-conorms are different. All non-symmetrical uninorms are not iso-morphic with any balanced t-conorm. All sets of uninorms related to balancedt-conorms and the set of non-symmetrical uninorms partition the family of alluninorms, i.e. they create equivalence classes of an equivalence relation, cf. [6].

Proposition 5. Let U = {u : u is a uninorm}. Let us consider balanced isomorphicmappings as defined in Remark 1. Then, the pair (U, %S) is an equivalencerelation if for every two uninorms u and v, u %S v iff u and v are isomorphic with

the same balanced t-conorm S or none of u and v is isomorphic with any balancedt-conorm S.

The same notes concern balanced t-norms and nullnorms. Namely:

Proposition 6. Let V = {v: v is a null norm}, then the pair (V, %T) i s a nequivalence relation if for every two nullnorms u and v, u %T v iff u and v areisomorphic with the same balanced t-norm T in its normal form or none of u and v

is isomorphic with any balanced t-norm T, cf. Remark 1.

These propositions characterize the family of all t-conorms (t-norms in nor-mal form) as a set of equivalence classes of the relation %S (%T, respectively).This relation is defined on the set of all uninorms (nullnorms, respectively).

On the other hand neither balanced triangular norms, nor uninorms, nornullnorms satisfy the preservation rule. Employment of the method of



27/40

balanced extension to uninorms leads to intuitively clear definition of balanceduninorms described as follow:

Definition 9. The balanced uninorm is a function:

U : 1; 1 1; 1 ! 1; 1

satisfying the following axioms:

The structure of balanced uninorms is displayed in Fig. 9. As in the case ofbalanced triangular norms, the values of balanced uninorms are determined bythe values of the uninorm and the symmetry principle in the squares [0,1] [0,1]and [1,0] [1,0]. The values of the balanced uninorm in the squares[0,1] [1, 0] and [1,0] [0,1] are unconstrained and could be defined accord-ing to subjective aim of application (this topic is not investigated here).

Obviously, similar considerations are valid in the case of nullnorms, thoughthe values of balanced nullnorms in the unconstrained area meet a differenttype of border conditions.

The convergence experiment gives the following results for a given balanceduninorm (assuming that the original uninorm is composed of the t-norm andthe t-conorm both being strong):

1. U(a, U(b, c)) = U(U(a, b), c) Associativity2. U(a, b) = U(b, a) Commutativity3. U(a, b) 6 U(c, d) if a 6 c and b 6 d Monotonicity

4. ($

e 2 [0, 1]) ("

x 2 [0,1]) U(x, e) = x Boundary conditions5. U(x,y) = N(U(N(x), N(y))) Symmetry

-1

-e

1

1

0

squizzedt-norm

squizzedt-conorm

betweenminandmax

betweenminandmax

e

e-1

squizzed

t-norm

squizzedt-conorm

betweenminandmax

betweenmin

andmax

-e

unconstrainedarea

-1

-a

1

1

0

squizzedt-conorm

squizzedt-norm

constantvalue

a

a

a-1

squizzed

t-conorm

squizzedt-norm

-a

unconstrainedarea

constantvalue

a

unconstrainedarea

unconstrainedarea

constantvalue

-a

constant

value-a

Balanced uninorm Balanced null norm

Fig. 9. The structures of balanced uninorms and balanced nullnorms.



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lim UUU. . . Ua; a . . .; a; a|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n times

!n!1

1 if e < a 6 1

0 if e < a < e

1 if 1 6 a < e

e if a e

8>>>>>>>:what is consistent with the preservation rule.

3.2. Hierarchies of balanced operators

Balanced triangular norms defined in Section 2.2 are isomorphic with unin-

orms and nullnorms. Therefore, the method of creation of balanced uninorms(i.e. immersion of classical uninorms in the extended space of fuzzy sets) couldbe replaced by applying the balanced extension process twice to t-conorms. Thefirst step creates the balanced t-conorm, then after isomorphic transforma-tion of the balanced t-conorm to the unipolar scale the second step createsthe balanced uninorm. Consequently, a balanced uninorm is a kind of a bal-anced t-conorm of the higher rank. This process could be continued creatingnext ranks of balanced operators. Finally, balanced triangular norms, unin-orms and nullnorms are products of the same process of the iterative balanced

extension method applied to classical triangular norms. This property explainssimilarity between balanced triangular norms, on one hand, and uninorms andnullnorms, on the other. The process of consecutive employment of the bal-anced extension method creates a hierarchy of balanced triangular norms.

Iterative t-conorm. A helper function, the so-called iterative t-conorm, isused as an illustration of creation of the balanced hierarchy.

Definition 10. For a given balanced t-conorm S the iterative t-conorm is afunction: S_iter : R R ! R

S iterx;y

Sx 2k 2l;y 2k 2l 2l x 2k 2l;y 2k 2l2 1;11;1 and k;l-integers

1 2l x 2k 2l;y 2k 2l 2 1;3

1;1 and k;l-integers

8>>>>>:

Note: The t-conorm S may vary for different areas of the domain. Thus, inthis case, the formula looks like:

S iterx;y

Sk;lx 2k 2l;y 2k 2l 2l x 2k 2l;y 2k 2l 2 1;1

1;1 and k;l-integers1 2l x 2k 2l;y 2k 2l 2 1;3

1;1 and k;l-integers

8>>>>>:where Sk,l is a balanced t-conorm for any values of k and l.



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Properties of the iterative t-conorm are determined by balanced t-con-orms Sk,l. For instance, continuity of the iterative t-conorm S_iter is deter-

mined by continuity of balanced t-conorms Sk,l. The iterative t-conormS_iter is non-continuous in all non-continuity points of balanced t-conormsSk,l. It also may be non-continuous on the borders of the upper-left andbottom-right quarters of the domain squares growing the balanced t-con-orm Sk,l. What is important is that the iterative t-conorm S_iter is alwaysnon-continuous in the upper-right and bottom-left vertexes of thesesquares of its domain where it is constant. It may be continuous everywhereelse.

Example: since the balanced t-norm S, based on the additive generator

fS(x) = x/(1 jxj), is non-continuous in the upper-left and bottom-right ver-texes of its domain, the respective iterative t-conorm S_iter is also non-contin-uous in all such points. Specifically, the iterative t-conorm S_iter based onthe additive generator fS(x) = x/(1 jxj) is a continuous function in its domainexcept for the following set of points {(1 2k 2l, 1 + 2k 2l) : k, l-integervalues}, i.e. except for the upper-right and bottom-left vertexes of thesesquares of its domain where it is constant. The contour plot of the iterativet-conorm, based on the additive generator fS(x) = x/(1 jxj), is outlined inFig. 10.

The hierarchy of balanced operators. Hierarchies of balanced t-conorms andbalanced t-norms are created based on an iterative t-conorm. The helper iter-ative t-conorm is a function defined on the whole plain R2 while hierarchies ofbalanced operators are fuzzy connectives with the domain [1,1] [1,1] andthe co-domain [1,1]. Balanced operators are parts of iterative t-normsqueezed to fit the square [1,1] [1,1].

Definition 11. The balanced t-conorm of rank n is the mapping

S

n:

1; 1 1; 1 ! 1; 1defined as a composition of three functions:

the function (tx, ty) of the linear rescaling of the square [1,1] [1,1] ontothe square [2k+ 2l 1, 2k+ 2l 1 + 2n] [2k+ 2l 1, 2k+ 2l 1 +2n] for some integers k and l;

the iterative t-conorm restricted to the square [2k+ 2l 1, 2k+ 2l 1 +2n] [2k+ 2l 1, 2k+ 2l 1 + 2n];

the function t1 of linear rescaling of the interval [2l 1, 2l 1 + 2n] onto

the interval [1,1].

Namely:

Sn t1 S iter tx; ty



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where for some integer numbers kand land a given positive integer number n:

tx : 1; 1 ! 2k 2l 1; 2k 2l 1 2ntxx nx 1 2k 2l 1

ty : 1; 1 ! 2k 2l 1;2k 2l 1 2n

tyx nx 1 2k 2l 1

t1

: 2l 1; 2l 1 2n ! 1; 1t1x x 2l 1=n 1

Definition 12. The balanced t-norm of rank n is the mapping:

Tn : 1; 1 1; 1 ! 1; 1

defined as a composition of three functions:

Fig. 10. The plot of iterative t-conorm based on the additive generator fS(x) = x/(1 jxj).



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the function (tx, ty) of the linear rescaling of the square [1,1] [1,1] ontothe square [2k+ 2l, 2k+ 2l+ 2n] [2k+ 2l, 2k+ 2l+ 2n] for some inte-

ger numbers k and l; the iterative t-conorm restricted to the square [2k+ 2l, 2k+ 2l+ 2n]

[2k+ 2l, 2k+ 2l+ 2n]; the function t1 of linear rescaling of the interval [2l, 2l+ 2n] onto the inter-

val [1,1].Namely:

Tn t1 S iter tx; ty

where for some integer numbers kand land given positive integer number n:

tx : 1; 1 ! 2k 2l; 2k 2l 2ntxx nx 1 2k 2l

ty : 1; 1 ! 2k 2l;2k 2l 2ntyx nx 1 2k 2l

t1 : 2l; 2l 2n ! 1; 1t1x x 2l=n 1

Figs. 11 and 12 illustrate the process of creation of the hierarchy of balancedt-norms and balanced t-conorms based on the iterative t-conorm. Because thebalanced t-norm and the balanced t-conorm of any given rank have the square[1,1] [1,1] as their domain, then a part of the iterative triangular normdefined by respective squares displayed in Figs. 11 and 12 must be transformedin order to fit this square.

For instance, the balanced t-conorm of rank 2, described by the part of theiterative triangular norm restricted to the square [5, 1] [1,3], is describedas follow. The mapping:

fun : 5;1 1; 3 ! 3; 1

funx;y S iterx;y

must be transformed using the transformation:

tx : 1; 1 ! 5;1; txx 2x 3 and

ty : 1; 1 ! 1; 3; tyy 2y 1 and

t1 : 3; 1 ! 1; 1; t1x x 1=2

which means that the balanced t-conorm of rank 2 S(2) corresponding to themapping fun is defined as follow:



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S2 : 1;11;1 ! 1;1

S2x;y t1 S iter tx; tyx;y t1S itertxx; tyy t

1funtxx; tyy

In other words, the graph of the mapping fun restricted to the cube [5,1] [1, 3] [3, 1] has to be squeezed to the cube [1,1] [1,1] [1,1] in order to create the balanced t-conorm of rank 2.

In Fig. 11 the balanced t-conorm of rank 2 is marked as a uninorm whatshould be interpreted as a relation between balanced t-conorms and uninormsin terms of Section 3.1. On the other hand, slightly modified iterative t-conormscould be used for the creation a hierarchy of balanced operators including alluninorms and nullnorms. This issue, as a subject for potential investigation, is

out of the scope of the paper and it will not be developed here.Uniform iterative norms. The iterative t-conorms (helper functions used in

definition of the hierarchy of balanced t-conorms and t-norms of given ranks)have plain areas, i.e. they are constant in half of their domains. To avoid plainareas other type of iterative operators are defined and then used in the defini-

Fig. 11. The hierarchy of balanced t-conorms.



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tion of the hierarchy of balanced operators. These operators are called uniformoperators.

Definition 13. The uniform iterative t-conorm is a mapping:S iter: R R !R

S iterx;y Sx 2k;y 2l 2k 2l wherex 2k;y 2l 2 1;1 1;1

and k; l-integers

where S is a balanced t-conorm.

Definition 14. The uniform iterative t-norm is a mapping:

T iter:

R R ! 1;1

T iterx;y Tx 2k;y 2k wherex 2k;y 2k 2 1;11;1

and k;l-integers

where T is a balanced t-norm in the weak inertia form.

Fig. 12. The hierarchy of balanced t-norms.



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The uniform iterative t-conorm S_iter and the uniform iterative t-norm T_iter may differ for different areas of the domain. Thus, the formu-

las may be rewritten in the spirit of the second formula of Definition10.

Examples of uniform iterative triangular norms based on the additivegenerator fS(x) = x/(1 jxj) are presented in Figs. 13 and 14. The indicatedparts of the uniform iterative t-conorm and of the uniform iterative t-normtransformed to the cube [1,1] [1,1] [1,1] will create uniform balancedt-conorms and t-norms of ranks 15. The transformation is similar to thatdeveloped in Definition 11. All these mappings are continuous in theirdomains except for isolated points corresponding to non-continuity points of

the basic t-conorm and t-norm used in definition of iterative operators. Letus recall that the basic t-conorm and t-norm generated by the additive gener-ator fS(x) = x/(1 jxj) are continuous in the set [1,1] [1,1] {(1,1),(1, 1)}.

Fig. 13. The structure of uniform iterative t-conorms.



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4. Final remarks

This paper is a contribution to a discussion on negativity and symmetry ofuncertain information processing. Motivation to this study comes from anobservation that the nature of fuzzy connectives such as t-norms and t-con-orms is inconsistent with Zadehs extension of crisp sets to fuzzy sets. Thisinconsistency is raised by the asymmetry of fuzzy sets, which define the gradeof inclusion of an element into a set while the grade of exclusion is not repre-sented. An attempt to endow the space of fuzzy sets with negative information(and to provide symmetry in this way) leads to balanced operators, which gen-

eralize the concept of t-norms and t-conorms. The idea of balanced t-norms inhe weak form expands the classical fuzzy connectives removing insensitivity ofbalanced t-norms and nullnorms. Two different implementations of balancedt-norms in the weak form were proposed: balanced t-norms in the weak inertiaform and balanced t-norms in the weak intuitive form.

Fig. 14. The structure of uniform iterative t-norm in weak inertia form.



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A broader range of iterative operators, as an outcome of balanced extension,is explored. The iterative operators create a hierarchy of operators with classical

and balanced operators at the bottom. Based on balanced t-conorms and t-norms in the weak inertia form a hierarchy of the so-called uniform iterativet-conorms and uniform iterative t-norms are proposed. The discussion is limitedto the presentation of basic concepts of introduced operators. Investigation andfurther discussion on formal properties of introduced operators is needed.

The directions of further research are related to practice and theory. Thepractical aspects of the balanced modeling could find its application in the deci-sion-making process with uncertain premises. Theoretical investigations wouldconcern formal characterization of the class of balanced t-conorms and t-

norms in the weak form and the class of uniform iterative norms in terms ofmathematical structures, investigation of dual families of balanced t-conormsand t-norms in the weak form, applications of newly introduced operators.

Acknowledgements

I am greatly indebted to Professor Witold Pedrycz for his relentless encour-agement to my work on this subject and support. I would also like to thankProfessor Radko Mesiar for his remarks to the early draft of the paper, com-ments on mistakes and weaknesses, directing to respective references and forhis encouragement.

Appendix A

Conclusion 7. The balanced t-norm defined in Propositions 3 and 4 satisfy thebalanced De Morgan association formula.

Proof. For (x,y) 2 [0,1] [0,1] association formulas are direct conclusion fromaxiomatic definitions of balanced triangular norms, for (x,y) 2 [1,0] [1,0]it also directly comes from axiomatic definitions of balanced triangular normsincluding symmetry property. T_norm defined in Propositions 3 and 4 obviouslysatisfies Definition 6 in its whole domain. So, the only condition to be shown is:

jTx;yj 6 jISIx;Iyj for x;y 2 1; 0 0; 1 [ 0; 1 1; 0

that could be reduced the following formula

ISSIjxj;Ijyj;IjSx;yj 6 jISIx;Iyj

Since I(jS(x,y)j)P 0 and S(I(jxj), I(jyj))P 0 the following inequality hold:

SSIjxj;Ijyj;IjSx;yjP S0;IjSx;yj IjSx;yj



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Of course: jI(S(I(x), I(y)))j = I(jS(I(x), I(y))j) so we can rewrite the aboveformula:

SSIjxj;Ijyj;IjSx;yjP jSIx;Iyj

and, finally, this is equivalent to:

ISSIjxj;Ijyj;IjSx;yj 6 IjSIx;Iyj

Conclusion 8. The balanced t-norms defined in Propositions 3 and 4 areconsistent

jTx;yj 6 jSx;yj for x;y 2 1; 1 1; 1

Proof. Satisfaction of the condition in the squares [1,0] [1,0] [ [0,1] [0,1] is obvious.

Since the following inequality is satisfied in the squares [1,0] [0,1] [ [0,1] [1,0] (see proof to Conclusion 7):

SSIjxj;Ijyj;IjSx;yjP S0;IjSx;yj IjSx;yj

then

jTx;yj ISSIjxj;Ijyj;IjSx;yj6IIjSx;yjjSx;yj

Conclusion 9. Extension of basic triangular norms minimum and maximumcan be approximated by strong Archimedean norms. Let fn(x) = sign(x) *(jxjn/(1 jxjn)) be an additive generator of balanced t-conorm. The balancedt-conorm Sn can approximate max co-norm to arbitrary accuracy

lim Snx;y !n!1

maxx;y for 0 6 x and 0 6 y

Proof. Let fnx signxjxjn

1jxjn, then f1n x signx

jxj1jxj

1=n.

The S_norm is described by following formula:

Snx;y

xn

1 xn

yn

1 yn

1 xn

1 xn

yn

1 yn

0BB@

1CCA

1=n

The assumption: 0 6 x 6 y < 1 allows for simplifying next formulas without

lost generality.yn

1 yn6

xn

1 xn

yn

1 yn6 2

yn

1 yn

These inequalities are obvious due to the assumption 0 6 x 6 y < 1.



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These inequalities imply the following relations:

12

1=ny

12

yn 1=n

6

yn

1 yn

1 2 yn

1 yn

0BB@ 1CCA1=n

6 Snx;y

6

2 yn

1 yn

1 yn

1 yn

0BB@

1CCA

1=n

2 yn1=n 21=n y

Since

lim1

2

1=n!

n!11 and lim 21=n !

n!11

we have

lim Snx;y !n!1

y

and finally we can conclude that for 0 6 x and 0 6 y:

lim Snx;y !n!1

maxx;y

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